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Which has a larger area

 

A triangle with dimensions 29 cm by 29 cm by 40 cm, or a triangle with dimensions 29 cm by 29 cm  by 42 cm

 

Pls explain solution 

 

Thanks

Guest Feb 12, 2016

Best Answer 

 #3
avatar+78557 
+10

They have exactly the same area......believe it, or not.....!!!!!

 

Using Heron's formula to prove this, we have, for the 29,29,40 triangle

 

sqrt [49(20)(20)(9) ] =  about 420 cm^2

 

And for the 29,29, 42 triangle, we have

 

sqrt [ (50(21)(21)(8) ]  =  420 cm^2

 

Here is the explanation for the derivation of Heron's Formula along with a calculator to verify the above results :

 

http://www.mathsisfun.com/geometry/herons-formula.html

 

 

 

cool cool cool

CPhill  Feb 12, 2016
edited by CPhill  Feb 12, 2016
edited by CPhill  Feb 12, 2016
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6+0 Answers

 #1
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0

These are isosceles triangles. Area=1/2 X base X height

So, one with base 42 would have a larger area.

Guest Feb 12, 2016
 #2
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+5

.....but the HEIGHT changes with the bigger BASE resulting in BOTH triangles having the same area. 

Guest Feb 12, 2016
 #3
avatar+78557 
+10
Best Answer

They have exactly the same area......believe it, or not.....!!!!!

 

Using Heron's formula to prove this, we have, for the 29,29,40 triangle

 

sqrt [49(20)(20)(9) ] =  about 420 cm^2

 

And for the 29,29, 42 triangle, we have

 

sqrt [ (50(21)(21)(8) ]  =  420 cm^2

 

Here is the explanation for the derivation of Heron's Formula along with a calculator to verify the above results :

 

http://www.mathsisfun.com/geometry/herons-formula.html

 

 

 

cool cool cool

CPhill  Feb 12, 2016
edited by CPhill  Feb 12, 2016
edited by CPhill  Feb 12, 2016
 #4
avatar+78557 
+10

Here's the visual proof of this.....

 

 

Draw a circle centered at the origin with a radius of 29  with the specified triangles inscribed in the bottom half of this circle

 

Triangle ECD  is isosceles with sides = 29, a base of 42  and a height, GC = 20

 

Triangle  BCA  is iscosceles with sides = 29, a base of 40  and a height,  FC = 21

 

So....area of ECD = [42 * 20]/2  =  840/2 = 420 cm^2

 

And BCA has an area of [ 40 * 21] / 2  = 840/2  =  420 cm^2

 

Thus   area of ECD  = area of BCA   ....!!!!

 

 

 

cool cool cool

CPhill  Feb 12, 2016
edited by CPhill  Feb 12, 2016
 #5
avatar+90970 
0

Thanks Chris,

That is a very unexpected outcome :))

Melody  Feb 14, 2016
 #6
avatar+78557 
0

One more comment about this problem....we will always have integer solutions and equal triangles when the quantties involved form Pythagorean Triples.......specifically......

 

If the isoceles sides of the triangles  =   "c"  in the formula  a^2 + b^2  = c^2

 

And if we let the base of one  triangle  = two times the length of one of the legs in the triple.....then the height will be equal to the remaining number in the triple.

 

 

cool cool cool

CPhill  Feb 14, 2016

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